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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Sequences and Series

NDA 2024 · Mathematics · Sequences and Series

Let p=In(x), q= (x³) and r= (x^5) , where x > 1 . Which of the...

Let p=In⁡(x),q=ln⁡(x3)p=\operatorname{In}(x), q=\ln \left(x³\right) and \mathrm{r}=\ln \left(x^5\right),where, wherex > 1.Whichofthefollowingstatementsis/arecorrect?A.. Which of the following statements is/are correct? A.p, q and rr are in AP. B. p,qp, q and rr can never be in GP. Select the answer using the code given below:

Options

  1. AA only
  2. BB only
  3. CBoth A and B
  4. DNeither A nor B

Correct answer

C. Both A and B

Step-by-step solution

We have given, p=ℓnx,q=ℓnx3=3ℓnxr=ℓln⁡x5=5ln⁡x\mathrm{p}=\ell \mathrm{n} \mathrm{x}, \mathrm{q}=\ell \mathrm{n} \mathrm{x}^3=3 \ell \mathrm{n} \mathrm{x}\mathrm{r}=\ell \ln \mathrm{x}^5=5 \ln \mathrm{x} Clearly, q−\mathrm{q}-\mathrm{p}=\mathrm{r}−q=2-\mathrm{q}=2 \ln x ⇒p,q\Rightarrow \mathrm{p}, \mathrm{q} and r are in A.P. Also, since qp≠rq⇒p,q\frac{\mathrm{q}}{\mathrm{p}} \neq \frac{\mathrm{r}}{\mathrm{q}} \Rightarrow \mathrm{p}, \mathrm{q} and r can never be in GP.
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